A diffusive Fisher-KPP equation with free boundaries and time-periodic advections

We consider a reaction-diffusion-advection equation of the form: $u_t=u_{xx}-β(t)u_x+f(t,u)$ for $x\in (g(t),h(t))$, where $β(t)$ is a $T$-periodic function representing the intensity of the advection, $f(t,u)$ is a Fisher-KPP type of nonlinearity, $T$-periodic in $t$, $g(t)$ and $h(t)$ are two free boundaries satisfying Stefan conditions. This equation can be used to describe the population dynamics in time-periodic environment with advection. Its homogeneous version (that is, both $β$ and $f$ are independent of $t$) was recently studied by Gu, Lou and Zhou \cite{GLZ}. In this paper we consider the time-periodic case and study the long time behavior of the solutions. We show that a vanishing-spreading dichotomy result holds when $β$ is small; a vanishing-transition-virtual spreading trichotomy result holds when $β$ is a medium-sized function; all solutions vanish when $β$ is large. Here the partition of $β(t)$ is much more complicated than the case when $β$ is a real number, since it depends not only on the "size" $\barβ:= \frac{1}{T}\int_0^T β(t) dt$ of $β(t)$ but also on its "shape" $\tildeβ(t) := β(t) - \barβ$.

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